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Defining Bernoulli polynomials in \(\mathbb Z/p\mathbb Z\) (a generic regularity condition). (English) Zbl 0694.10014

The authors consider functional equations characterizing Bernoulli polynomials. They study conditions under which these equations characterize functions in prime fields \(\mathbb Z/p\mathbb Z\). One typical result states that the “interpolation equation” \[ F(x)=q^{m-1}[F(\tfrac{x}{q})+F(\tfrac{x+1}{q})+F(\tfrac{x+q-1}{q})], \] where \(q\) is a primitive root mod \(p\) and \(1\leq m\leq p-1\), together with the value of \(F(0)\) completely characterizes a function \(F: \mathbb Z/p\mathbb Z\to \mathbb Z/p\mathbb Z\) if and only if \((q^ m-1)B_ m\not\equiv 0\pmod p\), where \(B_ m\) denotes the \(m\)th Bernoulli number. Connections to the regularity of primes, to the \(p\)-divisibility of Fermat quotients, and to a Voronoi type congruence are observed.

MSC:

11B68 Bernoulli and Euler numbers and polynomials
11A15 Power residues, reciprocity
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